Thermodynamics of Isoradial Quivers and Hyperbolic 3-Manifolds

The BPS sector of $\mathcal{N}=2$, $4d$ toric quiver gauge theories, and its corresponding D6-D2-D0 branes on Calabi-Yau threefolds, have been previously studied using integrable lattice models such as the crystal melting model and the dimer model. The asymptotics of the BPS sector, in the large N limit, can be studied using the Mahler measure theory, \cite{Zah}. In this work, we consider the class of isoradial quivers and study their thermodynamical observables and phase structure. Building on our previous results, and using the relation between the Mahler measure and hyperbolic 3-manifolds, we propose a new approach in the asymptotic analysis of the isoradial quivers. As a result, we obtain the observables such as the BPS free energy, the BPS entropy density and growth rate of the isoradial quivers, as a function of the $R$-charges of the quiver and in terms of the hyperbolic volumes and the dilogarithm functions. The phase structure of the isoradial quivers is studied via the analysis of the BPS entropy density at critical $R$-charges and universal results for the phase structure in this class are obtained. Explicit results for the observables are obtained in some concrete examples of the isoradial quivers.